Originally posted by FabianFnas Two different great circles are only parallell at to places. You cannot place a square 9 point dot grid at any places and expect it to be crossed by three straight lines.
Originally posted by FabianFnas Do I have to? Isn't it the basis of non-euclidian geometry?
Well - no, I think not. Non-Euclidean geometry deals with parallel lines. If we take a square to be four point a, b, c and d with d(a,b)=d(b,c)=d(c,d)=d(d,a) (d(u,v) = distance between u and v, a metric) then the lines aren't necessarily parallel, although they are in Euclidean geometry. If that makes sense.
Of course, such a square (the nine-pointed one) may not exist, but as far as I can tell this is not obvious...
Originally posted by Swlabr Well - no, I think not. Non-Euclidean geometry deals with parallel lines. If we take a square to be four point a, b, c and d with d(a,b)=d(b,c)=d(c,d)=d(d,a) (d(u,v) = distance between u and v, a metric) then the lines aren't necessarily parallel, although they are in Euclidean geometry. If that makes sense.
Of course, such a square (the nine-pointed one) may not exist, but as far as I can tell this is not obvious...
Well, take a globe and put a mark at the crossings between the longitudes 0 degrees, 30 degrees and 60 degrees west, and further the latitudes 0 degrees, 30 degrees and 60 degrees north. In a flat surface, they would in an exact square, right? But look at the globe, are they in a perfect square? No. The distance between the northwest point and the northeast point is only half the distance (or so) compared with the dots at the equator. Can we agree that the dots are not a perfect square? Not even if the nine dots are symmetricly placed around the equator. Is it possible to make a perfect square on the surface of a sphere? I say it's obviously impossible.
We have to be very free in the definition of 'straight lines on a sphere' in order to find a solution to the problem presented in this thread.
Originally posted by FabianFnas Well, take a globe and put a mark at the crossings between the longitudes 0 degrees, 30 degrees and 60 degrees west, and further the latitudes 0 degrees, 30 degrees and 60 degrees north. In a flat surface, they would in an exact square, right? But look at the globe, are they in a perfect square? No. The distance between the northwest point and the northea aight lines on a sphere' in order to find a solution to the problem presented in this thread.
That is because you are trying to project a square from a flat onto a sphere, which is, I believe, impossible - there exists no distance-preserving projection from a flat surface to a sphere, and vice-versa. However, you can cover a sphere with 6 squares, each with an angle of 120 degrees at the corners.
Originally posted by Swlabr That is because you are trying to project a square from a flat onto a sphere, which is, I believe, impossible - there exists no distance-preserving projection from a flat surface to a sphere, and vice-versa. However, you can cover a sphere with 6 squares, each with an angle of 120 degrees at the corners.
Right you are. Therefore his problem has no solution.
Unless you use my solution of the 9 dots on a plane.
But then on the other hand one stroke is enough.
Originally posted by FabianFnas Right you are. Therefore his problem has no solution.
Unless you use my solution of the 9 dots on a plane.
But then on the other hand one stroke is enough.
"...Draw nine dots so that they are arranged in a 3x3 square..."
It doesn't say anything about being on the Euclidean plane... 😉
Nine spots with 3 straight lines
01 Apr '09 07:00
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